Now that I think about it there seems to be a casuality the use of the same word spectrum, without any relation of one concept to the other, but when I asked my linear algebra teacher if there was some relation between (eigenvectors/eigenvalues) and the fourier transform she told me yes, without any more info as to what was that relation.

inner product space, is the only thing I can find , but other than that I am not completely sure

Also nothing in mathematics is a coincidence :D new discoveries are always ready to be found, new connections ready to be made, from now perspectives!

Hi Tenshou,
I agree, but my question was pointing more to if there was or not any reason why the word spectrum or spectra to be used for these two apparently completly different things: the spectra of a linear operator, and the spectra obtained by a fourier transformation.

The use of the same word is pure coincidence? Historically speaking at least?

For an extreme case of an overuse of a word in Mathematics, look up the term "Normal", which has a lot of very different uses. Normal in normal subgroup, Normal Topological Space, Normal Line, Normal Subspace....

A linear, time-invariant operator can be represented in terms of its impulse response, which is a time-domain function ##h## which relates an input ##x## to the corresponding ##y## by means of a convolution:
$$y(t) = \int_{-\infty}^{\infty} h(\tau) x(t - \tau) d\tau$$
If we choose ##x## to be a complex exponential function, of the form ##x(t) = a e^{i \omega t}##, then we obtain the result
$$y(t) = \int_{-\infty}^{\infty} h(\tau) a e^{i \omega (t - \tau)} d\tau = a e^{i \omega t} \int_{-\infty}^{\infty} h(\tau) e^{-i \omega \tau} d\tau = a e^{i \omega t} \hat{h}(\omega)$$
where ##\hat{h}## is the Fourier transform of ##h##. (Assuming that ##h## is integrable.) Thus, we have shown that a complex exponential is an eigenfunction of the operator corresponding to ##h##: if we apply the operator to a complex exponential, the result is the same complex exponential, scaled by the factor ##\hat{h}(\omega)##. Thus we may consider ##\hat{h}(\omega)## to be the eigenvalue associated with the eigenfunction ##e^{i\omega t}##. In Fourier analysis, we may define the spectrum to be the function ##\hat{h}##. The image of this function is simply the set of all eigenvalues, which corresponds to the definition of the spectrum in functional analysis.